Black 1 doesn't often work well to attack the two marked stones directly - but then nothing else which fills White's liberties will generally work much better.
White at the point 1 would improve White's liberties to six: other solid connections leave White with at most five.
Therefore if there is a tesuji round these stones, the nose play Black 1 is the best candidate.
Beginners do tend to play at the 'nose' far too often - their reasoning would be correct, if it were a question of liberties alone. This is part of the tendency to play too close.
Yamabe Toshiro-Sakata Eio 1967-02-08, from the Meijin League. White 1 here functions as a kind of inducing move. Black's previous play was at the marked stone. Having cut in the centre, White's problem is how to handle the cutting stones and the large weak group at the same time.
White does succeed in dealing with both areas of immediate concern. After Black 10 there followed White a, Black b, White c. At that point White had enough threats against Black 10 and the marked black stone to fight on, locally; and had isolated Black's group on the right side.